Waves in fluids and solids Part 5 docx

All the layers may be arbitrary piezoelectric, dielectric, metal, but if the layer is used as an electrode, its transfer matrix differs from matrices, described above.. It means that the

The factors sequence must be namely such, as in (64), any transposition is impossible in

general case, because A.B ≠ B.A for a matrices multiplication in general case The matrix M

in (64) transfers the values u j , T 1j , D1 and from the surface x1 = 0 (bottom) to the surface x1

= l1+l2+…+lN (top)

All the layers may be arbitrary (piezoelectric, dielectric, metal), but if the layer is used as an

electrode, its transfer matrix differs from matrices, described above It is obviously, that only

the metal layer can be used as an electrode Therefore all the mechanical values and the

electric potential of the electrode are transferred by the matrix (63) If the metal layer is not

connected to the electric source and is electrically neutral, the matrix (63) transfer the normal

component of the electric displacement correctly too, i.e (D1)x1=l = (D1)x1=0 (but not inside the

metal layer, where D1 = 0) But if the metal layer is connected to the electric source and is

used as an electrode, a discontinuity of the value D1 takes place which is not represented in

the matrix (63)

Therefore the special consideration is needed for electrodes Fig 6 shows two electrodes,

connected to an external harmonic voltage source with an amplitude V and a frequency 

Fig 6 Two electrodes, connected to an external harmonic voltage source with amplitude V

and frequency 

First we will consider electrodes of zero thickness Therefore all the mechanical values are

transferred without changes (electric potential is transferred without changes always by

metal layer of any thickness)

Values D1(1-) and D1(1+) on both sides of the first electrode are different, for the second

electrode analogously The difference D1(1+) - D1(1-) is equal to the electric charge per unit

area of the electrode (in the SI system) A time derivative of this value is the current density

Its multiplication on the electrode area A gives the total electrode current For a harmonic

signal the time derivative equivalent to a multiplication on i As a result the following

expression takes place for a current I1 of the electrode 1:

For electrode 2 analogously If there are only two electrodes connected to one electric source,

then I = I1 = - I2 and:

I = VY, (66)

where V = – (1 and 2 are electrode potentials) and Y is an admittance of two

electrodes for the external electric source

We are free in determining the zero point of the electric potential and we can choose it so:

As a result, we can obtain from (65) and (66):

which expresses the value of D1 at the upper side of the electrode as a linear function of the

values of and D1 at the lower side (has the same value on both sides of an electrode) It

means that the transfer matrix of the electrode of zero thickness (an ideal electrode) has a

The metal electrode of a finite thickness (a real electrode) can be presented as a combination

of two layers, one of which is the metal electrode of a zero thickness (an ideal electrode),

transferring only electric values, and another one is a layer of a finite thickness, transferring

only the mechanical values (mechanical layer) - see Fig 7

Fig 7 Representation of a real electrode as a combination of an ideal electrode and a

mechanical layer

Therefore we can obtain the whole transfer matrix of the real electrode as a multiplication of

a matrix of the ideal electrode (69) and a matrix, transferring only mechanical values and

presented by expression (63):

As it was mentioned above, the matrices don’t obey the commutative law in general case,

but in this concrete case one can transpose these two matrices, what can be checked by

direct multiplication This means, in particular, that an ideal electrode can be placed on any

side of the read electrode, as shown in Fig 7 Physically more correctly to place the ideal

electrode on the side which is a face of contact with the interelectrode space

As a result, the multilayer bulk acoustic wave resonator, containing arbitrary quantity of

arbitrary layers, but only with two electrodes, has a view, presented in Fig 8

Fig 8 Multilayer bulk acoustic wave resonator with two electrodes

Here F is a combination of arbitrary quantity of arbitrary layers under electrodes, G is the same above electrodes, Q is the same between electrodes (at least one of layers in Q must be piezoelectric), E1 and E2 are the two electrodes of a finite thickness

All the eight values u j , T 1j , D1 and are transferring from a lower surface of the whole construction to its upper surface by the whole transfer matrix, which is the multiplication of transfer matrices of each elements:

Transfer matrices MF, MQ, MG are calculated by (64) and matrices M E1 and ME2 – by (70)

Because of electrodes presence the total transfer matrix of the whole resonator MFE1QE2G does not have generally the special form with 0 and 1 in the 7th column and the 8th row (as in (57)), but it is of the most general form:

The expressions, obtained above, allow to calculate the admittance of the resonator Y which

is its main work characteristic

The zero boundary conditions for T 1j and D1 on the external free lower and upper surfaces

of the construction are used for these calculations:

T11 = 0, T12 = 0, T13 = 0, D1 = 0 on free surfaces (73) The normal components of a stress tensor are equal to zero because lower and upper surfaces are free, the electric displacement is zero because the electric field of the external source is concentrated only between two electrodes (between their inner surfaces)

E2

F

G

Q

x1

E1

Let us denote the mechanical displacements and the electric potential on the lower free

u u u  Then with taking into account (73) these values will be connected each

other by the transfer matrix MFE1QE2G by the following expression:

(1) 3

We can obtain the vector u(1)1 ,u(1)2 ,u(1)3 from the first equation (75) (using the standard

inverse matrix designation):

1 (1)

2 (1)

This is the main equation of the problem It connects the resonator admittance Y with the

frequency  because Y value is contained in the transfer matrices of electrodes We can set

the concrete value of  and calculate from (78) the corresponding value of Y, i.e we can

obtain the frequency response of the resonator – its main work characteristic Matrix elements in (78) are elements of the total transfer matrix of the whole device – see (72)

In an arbitrary case the equation (78) cannot be solved analytically The solution can be found only by some numerical method We used our own algorithm of searching for the global extremum of a function of several variables (Dvoesherstov et al., 1999) Solution corresponds to the global minimum of the square of the absolute value of the left part of the equation (78) Two arguments of this function are the real and imaginary parts of the

admittance Y (for each given frequency)

If there is not any piezoelectric layer in the packets F and G outside the electrodes, the transfer matrices of these packets have the simpler form (62) and the equation (78) can be solved analytically in the following view:

M means the 1x3 matrix including the columns 4 – 6 and the 7th row of the 8x8 matrix,

and so on Index Q means that all these elements are taken from transfer matrix of the Q

packet (not for the whole device) "

In these expressions the lower indexes F and G also designate the corresponding packets,

M F and M G are the whole 8x8 matrices of the corresponding packets, M E1m and M E2m – the

“mechanical” parts of the electrodes 8x8 matrices and upper indexes uu, Tu, uT and TT

means that corresponding 3x3 matrices are taken from whole 8x8 matrices

Practically all the concrete FBAR devices do not contain piezoelectric layers outside the electrodes, i.e practically for all these devices the frequency response can be calculated with expressions (79) – (80)

But not only the frequency response can be calculated by the technique, described here The

expression (57) allows to calculate all eight values u j , T 1j , D1 and not only on the second

surface of the layer but also for any coordinate x1 inside the layer, if these eight values are known for the first “input” surface of this layer These values on the second “output” surface of the first layer can be used as “input” values for the second layer for the same

calculations for any coordinate x1 inside the second layer and so on, i.e the spatial distribution of all eight values inside the whole multilayer system can be obtained As was mentioned above, the values on the first “input” surface of the first layer must be known for such calculations (for frequency response calculations all eight values on the first surface of the first layer are not needed)

Four of eight values, namely, T 1j and D1 are known, they are zero – see (73) The absolute value of the electric potential is not essential from point of view of the spatial distribution of all eight values We can set any (but not zero) value of the electric potential on the first surface of the first layer, for instance (1) = -1 V Then we can obtain all three values of the mechanical displacements u(1)1 ,u(1)2 ,u(1)3 from the equation (76) So all eight values on the

first surface of the first layer are determined and the spatial distribution of all these values

can be obtained for any multilayer resonator with two electrodes The admittance Y for

given frequency  must be calculated preliminary, because both these values are needed for the spatial distribution calculation

The spatial distribution gives a possibility to obtain some information about physical wave processes those take place inside the multilayer structure, in particular - how the Bragg reflector “works”

Fig 9 shows the frequency response of the membrane type resonator (as in Fig 3a), obtained

by technique, described above The mass density of all the materials are taken in a form (1 + i), where  =-0.001 in this case The frequency response is calculated for two variants of the Al electrode thickness – zero and 0.1 m

a) b) Fig 9 Frequency response of the membrane type resonator Active layer – AlN, thickness 1

m a) – zero electrode thikness, Fres = 5.337 GHz, b) – Al electrode thickness 0.1 m, Fres = 4.577 GHz Electrode area 0.01 mm2

Fig 9 illustrates an influence of the electrode thickness on a resonance frequency (this frequency is obtained directly from a graphic as coordinate of a maximum of a Y real part) The resonance frequency is decreased by the electrodes of a finite thickness, because the whole device with more total thickness corresponds to more half-wavelength This

illustrates Fig 10 in which the spatial distribution of the T11 component of the stress tensor is shown, obtained also by a technique, described above

a) F = Fres = 5.337 GHz b) F = Fres = 4.577 GHz

Fig 10 Spatial distribution of T11 component of the stress tensor for two variants, shown in Fig 9 F = Fres in both cases

A half-wavelength corresponds to a distance between neighbouring points with zero stress

In a case a) this distance is 1 m and corresponds to a resonance frequency 5.337 GHz, whereas in a case b) a half-wavelength is equal to 1.2 m and corresponds to a lower frequency 4.577 GHz This gives a possibility to control the resonance frequency by changing of the top electrode thickness For example, Fig 11 shows dependences of the resonance frequency on a top electrode thickness for two materials of this electrode – Al and

Au The bottom electrode is Al of a thickness 0.1 m in both cases

Fig 11 Dependences of the resonance frequency on the top electrode tickness for Al and Au The bottom electrode is Al (0.1 m) in both cases The thickness of AlN is 1 m

For displaying of the possibilities of the described method Fig 12 shows also the spatial

distributions of the longitudinal component of the displacement u1 and the electric potential

 for the membrane type resonator, corresponding to Figs 9b and 10b

a) b)

Fig 12 Spatial distribution of the longitudinal component of the displacement u1 (a) and the electric potential  (b) for the membrane type resonator with Al electrodes of finite thickness 0.1 m

Distribution of D1 is not shown here because it is very simple – D1 = const between the electrodes and equal to zero outside the inner surfaces of the electrodes

If membrane type resonator is placed on the substrate of not very large thickness, then multiple modes appear, and this resonator can be a multi-frequency resonator, as shown in Fig 13a

a) b) Fig 13 FBAR membrane type resonator on a Si substrate of thicness 100 m (a) and 1000 m (b) Electrodes – Al, thicness 0.1 m, active layer – AlN, thickness 1 m

But if the substrate is too thick, there are too many modes and the resonator transforms from multi-mode actually into a “not any mode” resonator, as one can see in Fig 13b

So, the membrane type resonator cannot be placed on the massive substrate directly because

of an acoustic interaction with this substrate One must to provide an acoustic isolation between an active zone of the resonator and a substrate One of techniques of such isolation

is a Bragg reflector between the active zone and the substrate (as shown in Fig 3b) This reflector contains several pairs of materials with different acoustic properties The difference

of the acoustic properties of two materials in a pair must not be small Acoustic properties of materials, used for Bragg reflector, are characterized by a value V, where  is a mass

density and V is a velocity Values V are shown in Fig 14 for some isotropic materials

Material constants are taken from (Ballandras et al., 1997)

Fig 14 The value V for some isotropic materials

As one can see in Fig 14, the best combination for a Bragg reflector is SiO2/W A pair Ti/W

is good too, and a combination Ti/Mo also can be used successfully (combinations of Au or

Pt with other materials also can be not bad, but not cheap)

The thickness of each layer of the reflector must be equal to a quarter-wavelength in a material of the layer for a resonance frequency As it was mentioned above, the resonance frequency is defined mainly by the active layer thickness and can be adjusted by proper choice of the top electrode thickness

The computation technique, based on the described here rigorous solution of the wave

equations, allows to calculate any bulk wave resonators with any quantity of any layers,

including the resonators with Bragg reflector For example, Fig 15a shows a frequency

response of the resonator, containing an AlN active layer (1 m), two Al electrodes (both 0.2

m), three pairs of layers SiO2/W, and a Si substrate (1000 m)

a) b)

Fig 15 A frequency response (a) and a distribution of u1 (b) for a resonator with a Bragg

reflector, containing three pairs of layers SiO2/W

A thickness of a Bragg reflector layer is 0.38 m for SiO2 and 0.33 m for W (a

quarter-wavelength in a corresponding material for a resonance frequency) Fig 15a shows, that

three pairs of SiO2/W combination is quite enough for full acoustic isolation of an active

zone and a substrate A spatial distribution of a wave amplitude illustrates an influence of

the Bragg reflector on a wave propagation, for example, Fig 15b shows this distribution for

a longitudinal component of a mechanical displacement A coordinate axis x here is directed

from a top surface of a top electrode (x = 0) towards a substrate One can see in Fig 15b that

a wave rapidly attenuates in the Bragg reflector and does not reach the substrate

Calculation results show, that the first layer after an electrode must be one with lower value

V – the SiO2 layer in this case In a contrary case a reflection will not take place

If difference of values V of two layers of each pair is not large enough, then three pairs may

not be sufficient for effective reflection For example, calculations show that three or even

four pairs of Ti/Mo layers are not sufficient for suppressing the wave in the substrate Only

five pairs give a desired effect in this case and provide results similar shown in Fig 15 for

SiO2/W layers

So, the described technique allows to calculate any multilayer FBAR resonators, containing

any combinations of any quantity of any layers The main results of these calculations are a

frequency response of a resonator and spatial distributions of physical characteristics of the

wave (displacement, stress, electric displacement and potential)

In addition this technique gives a possibility to calculate a thermal sensitivity of the

resonator too, i.e an influence a temperature on a resonance frequency A resonance

frequency always changes in general case when a temperature changes This change is

characterized by a temperature coefficient of a frequency:

r

dF TCF

F dT

Here T is a temperature, F r is a resonance frequency

A computation technique, used here, allows to apply this expression for TCF calculation

directly and to calculate this value by numerical differentiation

A temperature influence on a resonance frequency is due to three basic causes:

1 A temperature dependence of material constants (stiffness, piezoelectric, dielectric

tensors) - TCFC

2 A temperature dependence of a mass density – TCF

3 A temperature dependence of a layer thickness – TCFh

A temperature dependence of material constants is described by temperature coefficients of these constants, a temperature dependence of a mass density is described by three linear expansion coefficients or by a single bulk expansion coefficient, a temperature dependence

of a thickness is described by a linear expansion coefficient along a thickness direction All these coefficients can be found in a literature, for example, for materials, usually used for FBAR resonators, one can see corresponding values in (Ivira et al., 2008)

First we will consider the simplest variant – a membrane type FBAR resonator with a single AlN layer and infinite thin electrodes For typical values of AlN temperature coefficients we can easily obtain:

TCF = TCFc + TCF + TCFh = (-29.639 +7.343 – 5.268).10-6/оС = -27.564.10-6/оС

One can check by a direct calculation, that this result does not depend on a thickness of AlN layer (for this variant with electrodes of finite thickness and for any multilayer structure

with layers of finite thickness it is not so) TCF value is always positive, TCFh value is

always negative A sign of TCFc is defined mainly by a sign of temperature coefficients of stiffness constants If temperature coefficients of stiffness constants are negative (for most

materials, including AlN), then TCFc is negative, if temperature coefficients of some stiffness

constants are positive (rare case, for example quartz), then TCFc can be positive and a total

TCF can be zero

For AlN a TCF value is always negative Al electrodes aggravate this position, besause

temperature coefficients of Al stiffness constants are negative too From this point of view

Mo electrodes are more preferable, because absolute values of temperature coefficients of its stiffness constants are significantly less than ones for Al (althouth they are also negative) For example, the concrete membrane type resonator Al/AlN/Al with an Al

thickness 0.2 m and an AlN thickness 1.1 m we can obtain: TCF = -44.23.10-6/оС (Fr =

3.648 GHz), and for Mo/AlN/Mo resonator with the same geometry: TCF = -33.76.10-6/оС

(Fr = 2.615 GHz)

For most applications a resonator must be thermostable, i.e TCF must be equal to zero The single possibility to compensate the negative TCF of AlN and of electrodes and to provide a total zero TCF is to add some additional layer with positive temperature coefficients of

stiffness constants Such material is, for example SiO2 Fig 16 shows dependenses of TCF of

membrane type resonator with Mo electrodes on a thickness ht of a SiO2 layer for two cases: SiO2 layer is placed together with AlN layer between electrodes (structure Mo/SiO2/AlN/Mo) and SiO2 layer is placed outside the electrodes (structure SiO2/Mo/AlN/Mo) Corresponding dependences of a resonance frequency are presented in Fig 16 too

Fig 16 shows that a SiO2 layer more effectively influences on both TCF and a resonance

frequency, when it is placed between electrodes

a) b)

Fig 16 Dependences of TCF and a resonance frequency on a SiO2 layer thickness ht for cases, when SiO2 is placed between electrodes (a) and when SiO2 is placed outside electrodes (b) A thickness of Mo electrodes is 0.06 m, a thickness of AlN is 1.9 m

Calculations show that a Bragg reflector does not change a resonance frequency of the corresponding membrane type resonator, if a thickness of each layer of the reflector is exactly

equal to a quarter-wavelength But a Bragg reflector influences on a TCF For this reason it is

reasonable to choose SiO2 as one material of a reflector In this case a thickness of an additional compensating SiO2 layer can be reduced For example, a thickness of SiO2 layer outside

electrodes, corresponding to TCF = 0, is equal about 0.53 m for variant, shown in Fig 16b for

membrane type resonator A resonance frequency is about 2.11 GHz for this case The Bragg reflector with three pairs of SiO2/Mo, corresponding this frequency, does not change this

frequency, but a TCF becomes positive due to SiO2 material presense in the reflector One must reduce a thickness of an additional compensating SiO2 layer to return a TCF to zero But then a resonance frequency will increase We must either increase an AlN layer thickness to return a resonance frequency or to change thickness of a Bragg reflector layers to adjust the reflector to

a new resonance frequency In any case several steps of sequential approximation are necessary The technique, described here, allows to do this without problem For example, presented in Fig 16b, full thermocompensation can be obtained for ht = 0.4 m (instead of 0.53

m for membrane type resonator) and for thickness of SiO2 and Mo layers in a Bragg reflector 0.71 m and 0.75 m respectively The AlN layer thickness remains 1.9 m and a resonance frequency slightly shifts remaining in the vicinity of 2.1 GHz

In many cases a presentation of FBAR resonator by means of some equivalent circuit is convenient – see for example (Hara et al., 2009) The simplest variant of an equivalent circuit is shown in Fig 17

Fig 17 An equivalent circuit of FBAR resonator

C0

Hear C0 is a static capacitance of a resonator – a real physical value, which can be calculated

by the geometry of the resonator and the dielectric properties of the layers between the

electrodes:

1 0

1 m i i i

l C

where i and l i is a relative dielectric permittivity (element 11 of a tensor) of a layer number i

and its thickness, 0 = 8.854.10-12 F/m – the dielectric constant, A is an area of a resonator

electrode, m is a quantity of layers between electrodes

Values Cm, Lm, and Rm are equivalent dinamic capacitance, inductance and resistance of the

resonator – values, which can not be determined from any physical representation – only by

comparison with experimental frequency response or with response, obtained by some exact

theory Theory, described here, allows to obtain these values

An admittance of the equivalent circuit, shown in Fig 17, can be calculated by following

m Rm

m

R Y

m m

m

L C

YRm and YIm are an active and reactive components of a dinamic admittance of the resonator,

jC 0 is an admittance of the static capacitance

Comparison of admittance, calculated by (83) and (84), with admittance, calculated by a

rigorous theory, described here, allows to obtain the unique values Cm, Lm, and Rm, which

give a frequency response, equivalent to the response, given by the rigorous theory

The resonance frequency of the equivalent circuit, shown in Fig 17, is defined as:

12

R Y

We can find a quality-factor from curve of a active component of the admittance:

r F Q F

